Bernstein components via the Bernstein center

Let G be a reductive p-adic group. Let \(\Phi \) be an invariant distribution on G lying in the Bernstein center \({\mathcal {Z}}(G)\). We prove that \(\Phi \) is supported on compact elements in G if and only if it defines a constant function on every component of the set \({\text {Irr}}(G)\); in particular, we show that the space of all elements of \({\mathcal {Z}}(G)\) supported on compact elements is a subalgebra of \({\mathcal {Z}}(G)\). Our proof is a slight modification of the argument from Section 2 of Dat (J Reine Angew Math 554:69–103, 2003), where our result is proved in one direction.     

Sobolev Error Estimates and a Bernstein Inequality for Scattered Data Interpolation via Radial Basis Functions

Error estimates for scattered-data interpolation via radial basis functions (RBFs) for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. Recently, these estimates have been extended to apply to certain classes of target functions generating the data which are outside the associated RKHS. However, these classes of functions still were not "large" enough to be applicable to a number of practical situations. In this paper we obtain Sobolev-type error estimates on compact regions of Rⁿ when the RBFs have Fourier transforms that decay algebraically. In addition, we derive a Bernstein inequality for spaces of finite shifts of an RBF in terms of the minimal separation parameter.     

Nilpotency in Bernstein algebras

This paper was written while the author held a grant from the CNPq of Brazil.     

Total Positivity in Stable Semigroups

We characterize the total positivity in space-time of strictly stable semigroups on \({\mathbb {R}}.\) In the positive case, this solves a problem that had been raised by Karlin. In the drifted Cauchy case, this concludes a study that we initiated in a previous paper. The case of the isotropic stable semigroup on \({\mathbb {R}}^d\) is also investigated. We apply these results to the bell-shape and monotone likelihood properties of certain stable densities.     

Positivity theorem in semi-group theory

We translate in semi-group theory the proof of the theorem of Ben-Arous and ourself got by the Malliavin Calculus giving a sufficient condition involved with the Bismutian distance such that an heat-kernel is strictly positive.     

Positivity aspects of the Fantappiè transform

A Hilbert space approach to the classical Fantappiè transform, based on the concept of Gel'fand triples of locally convex spaces, leads to a novel proof of Martineau-Aizenberg duality theorem. A study of Fantappiè transforms of positive measures on the unit ball inC ⁿ relates ideas of realization theory of multivariate linear systems, locally convex duality and pluripotential theory. This is applied to obtain von Neumann type estimates on the joint numerical range of tuples of Hilbert space operators. Partially supported by National Science Foundation Grants DMS 0070639 and DMS 0322255. Partially supported by National Science Foundation Grant DMS 0100367.     

Positivity of the K-entropy on non-abelian K-flows

A non-commutative extension of certain aspects of classical probability theory is presented in such a manner that the notion of Kolmogorov entropy can be extended to a large class of non-classical dynamical systems. In particular, the generalized K-entropy so defined is shown to be strictly positive on the class of non-abelian K-flows.     

On the affine Bernstein problem

A partial solution to the affine Bernstein problem is given. The elliptic paraboloid is characterized as a locally strongly convex, affine complete, affine-maximal surface in A ³, satisfying a certain growth condition, about its affine Gauss-Kronecker curvature.Research partially supported by DGICYT Grant PS87-0115-CO3-02.     

On localization of functions in the Bernstein space

For ϱ > 0, let be the closed subspace of L ¹(ℝ) consisting of functions ƒ having the Fourier transforms ƒ concentrated in [−ϱ, ϱ]. Let a > 0. In this paper, we consider the problem of maximal localization of the L ¹ norm on [−a, a] of functions from B _ϱ ¹ . More precisely, for a given a, we investigate the supremum of the quantity E _a (ƒ) = ∫ _−a ^a |ƒ(x)|dx over all ƒ from the unit ball of the space B _ϱ ¹ . __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 573–590, October–December, 2007.     

On the Positivity of the Stochastic Heat Equation

We study the positivity preserving properties of the heat equation with a white noise potential and random initial condition. Moreover, we find a generalized Feynman--Kac formula for the solution of the problem using methods from the white noise analysis. The initial condition can anticipate the driving white noise. We show that the solution is positive, when the random initial condition is positive. For the case of a time-dependent white noise potential, we give a special representation of the solution together with regularity results.     

New representation of the remainder in the Bernstein approximation

This short note presents a new representation of the remainder in the Bernstein approximation based on divided differences and some immediate applications. It is the only known representation of the remainder in the Bernstein approximation of arbitrary functions as a convex combination of divided differences of second order on known knots. As an application we obtain sharp inequalities for functions possessing bounded divided differences of second order and a new proof of the classical Weierstrass approximation theorem.     

Q_p空间中的Bernstein定理

最近,我们得到了单位圆盘Q_p空间中的Jackson定理,进一步建立其逆定理(Bernstein定理).为此,需要建立Q_p空间中的Bernstein不等式和Q_p空间范数的无导数特征刻画.后者的推导将利用Riesz插值公式,该公式将导数算子表示为平移算子.作为应用,给出了Q_p空间中的Lipschitz和Zygmund子空间的利用逼近表达的等价刻画.     

Positivity and Convexity in Rings of Fractions

Given a commutative ring A equipped with a preordering A⁺ (in the most general sense, see below), we look for a fractional ring extension (= “ring of quotients” in the sense of Lambek et al. [L]) as big as possible such that A⁺ extends to a preordering R⁺ of R (i.e. with A ∩ R⁺ = A⁺) in a natural way. We then ask for subextensions A ⊂ B of A ⊂ R such that A is convex in B with respect to B⁺ : = B ∩ R⁺. Supported by DFG. A short form of this article has been delivered at the conference Carthapos 2006 at Carthago (Tunisia).     

A Survey of Total Positivity

This paper contains an exposition of the classical work of Schoenberg and Gantmacher, Krein on the theory of totally positive matrices. It also contains a generalization of this theory to the Lie theoretic context and an application to the recent results of Fock and Goncharov on Teichmüller theory. Leonardo da Vinci lecture held on May 28, 2007 Received: June 2007